Variational theory and domain decomposition for nonlocal problems

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.     

Three dimensional dynamics of pretwisted beams: A spectral-Tchebychev solution

This paper presents the application of the spectral-Tchebychev (ST) technique for solution of three-dimensional dynamics of unconstrained pretwisted beams with general cross-section (including both straight and curved cross-sections). In general, the dynamic response of pretwisted beams presents three-dimensional (3D) motions, including coupled bending–bending–torsional–axial motions. As such, accurately solving pretwisted beam dynamics requires a 3D solution approach. In this work, the integral boundary value problem based on the 3D linear elasticity equations is solved numerically using the 3D-ST approach. To simplify evaluation of the volume integrals, the boundaries are simplified by applying two coordinate transformations to render the pretwisted beam with curved cross-section into an equivalent straight beam with rectangular cross-section. Three sample pretwisted beam problems with rectangular, curved, and airfoil cross-sections at different twist rates are solved using the presented approach. In each case, the convergence of the solution is analyzed, and non-dimensional natural frequencies and mode shapes are compared to those from a finite-element (FE) solution. Furthermore, cross-sectional stress and displacements are obtained from the 3D-ST solution. Lastly, the non-dimensional natural frequencies from the 3D-ST and a 1D/2D solutions are compared. It is concluded that the 3D-ST solution can capture the three-dimensional dynamic behavior of pretwisted beams as accurately as an FE solution, but for a fraction of the computational cost. Furthermore, it is shown that 1D/2D solution can lead to significant errors at high twist rates, and thus, the 3D-ST solution should be preferred.     

Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

We consider the problem of optimal portfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and where short positions are allowed. When the distribution of returns of risky assets is unknown but the mean return vector and variance/covariance matrix of the risky assets are fixed, we derive the distributionally robust portfolio rules. Then, we address uncertainty (ambiguity) in the mean return vector in addition to distribution ambiguity, and derive the optimal portfolio rules when the uncertainty in the return vector is modeled via an ellipsoidal uncertainty set. In the presence of a riskless asset, the robust CVaR and VaR measures, coupled with a minimum mean return constraint, yield simple, mean-variance efficient optimal portfolio rules. In a market without the riskless asset, we obtain a closed-form portfolio rule that generalizes earlier results, without a minimum mean return restriction.     

Atomic layer deposited HfO2 based metal insulator semiconductor GaN ultraviolet photodetectors

A report on GaN based metal insulator semiconductor (MIS) ultraviolet (UV) photodetectors (PDs) with atomic layer deposited (ALD) 5-nm-thick HfO₂ insulating layer is presented. Very low dark current of 2.24 × 10⁻¹¹ A and increased photo to dark current contrast ratio was achieved at 10 V. It was found that the dark current was drastically reduced by seven orders of magnitude at 10 V compared to samples without HfO₂ insulating layer. The observed decrease in dark current is attributed to the large barrier height which is due to introduction of HfO₂ insulating layer and the calculated barrier height was obtained as 0.95 eV. The peak responsivity of HfO₂ inserted device was 0.44 mA/W at bias voltage of 15 V.     

A Novel Thermotropic Elastomer based on Highly‐filled LDH‐SSB Composites

Elastomeric composites are prepared based on solution styrene butadiene elastomer and zinc‐aluminium layered double hydroxides (LDH), using a conventional sulphur cure system. Up to 100 parts per hundred rubber of LDH are incorporated into the elastomer matrix. The composites exhibit an interesting phenomenon of thermoreversible transparency, i.e. the transparent sample becomes opaque at warm condition and restores the transparency at room temperature. The transparency is found to be increased as the amount of LDH was increased. The addition of LDH gradually improved the mechanical, dynamic mechanical performance and thermal stability of the base elastomer. These developped elastomers could be utilised as smart materials in different applications.     

Theoretical study on the structures and stabilities of silacyclopropylidenoids

Isomeric structures, energies, and properties of silacyclopropylidenoids, C₂H₄SiMX (where M?=?Li or Na and X?=?F, Cl or Br), were studied ab initio at the HF and MP2 levels of theory using the 6-31+G(d,p) and aug-cc-pVTZ basis sets. The calculations indicate that each of C₂H₄SiMXs has three stationary structures: silacyclopropylidenoid (S), tetrahedral (T), and inverted (I). All of the silacyclopropylidenoid (S) forms are energetically more stable than others except that S-LiF is by only 0.7?kcal/mol higher in energy than I-LiF. In contrast, all of the tetrahedral (T) forms are the most unstable ones except for T-NaF. Energy differences between S, T, and I forms range from 0.70 to 8.70?kcal?mol^?1 at the MP2/6-31+G(d,p) level. In addition, the molecular electrostatic potential maps, natural bond orbitals, and frontier molecular orbitals were calculated at the MP2/6-31+G(d,p) level.     

Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α?>?0 such that ${|\Delta(E)| \gtrsim q}$ whenever ${|E| \gtrsim q^{\alpha}}$ , where ${E \subset {\mathbb {F}}_q^d}$ , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here ${\Delta(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x,y \in E\}}$ . Iosevich and Rudnev (Trans Am Math Soc 359(12):6127–6142, 2007) established the threshold ${\frac{d+1}{2}}$ , and in Hart et?al. (Trans Am Math Soc 363:3255–3275, 2011) proved that this exponent is sharp in odd dimensions. In two dimensions we improve the exponent to ${\tfrac{4}{3}}$ , consistent with the corresponding exponent in Euclidean space obtained by Wolff (Int Math Res Not 10:547–567, 1999). The pinned distance set ${\Delta_y(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x\in E\}}$ for a pin ${y\in E}$ has been studied in the Euclidean setting. Peres and Schlag (Duke Math J 102:193–251, 2000) showed that if the Hausdorff dimension of a set E is greater than ${\tfrac{d+1}{2}}$ , then the Lebesgue measure of Δ _y (E) is positive for almost every pin y. In this paper, we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set ${\Pi_y(E)=\{x\cdot y: x\in E\}}$ . Under the additional assumption that the set E has Cartesian product structure we improve the pinned threshold for both distances and dot products to ${\frac{d^2}{2d-1}}$ . The pinned dot product result for Cartesian products implies the following sum-product result. Let ${A\subset \mathbb F_q}$ and ${z\in \mathbb F^*_q}$ . If ${|A|\geq q^{\frac{d}{2d-1}}}$ then there exists a subset ${E'\subset A\times \dots \times A=A^{d-1}}$ with ${|E'|\gtrsim |A|^{d-1}}$ such that for any ${(a_1,\dots, a_{d-1}) \in E'}$ , $$ |a_1A+a_2A+\dots +a_{d-1}A+zA| > \frac{q}{2}$$ where ${a_j A=\{a_ja:a \in A\},j=1,\dots,d-1}$ . A generalization of the Falconer distance problem is to determine the minimal α?>?0 such that E contains a congruent copy of a positive proportion of k-simplices whenever ${|E| \gtrsim q^{\alpha}}$ . Here the authors improve on known results (for k?>?3) using Fourier analytic methods, showing that α may be taken to be ${\frac{d+k}{2}}$ .